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Results tagged with ap.analysis-of-pdes
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user 36721
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
8
votes
Accepted
Solutions and asymptotics of the ODE $ f''=f^{-\alpha} $
$\newcommand\al\alpha\newcommand\de\delta\newcommand\R{\Bbb R}\newcommand\B{\mathrm B}$It follows from the ODE
$$f''=f^{-\al} \tag{1}\label{10} $$
and the condition $f>0$ that $f$ is (strictly) convex …
- 128k
1
vote
Accepted
Domains of type (A) are Lipschitz?
$\newcommand\Om\Omega\newcommand\th\theta\newcommand\p\partial$After the previous answer was given, the OP changed "Lipschitz connected" to "uniform Lipschitz". If the latter is understood in the sens …
- 128k
1
vote
Domains of type (A) are Lipschitz?
A counterexample: $n=2$,
$$\Omega=B^-_{(0,0)}(2)\cup B^+_{(1,0)}(1)\cup B^+_{(-1,0)}(1),$$
where $B^\pm_C(r):=B_C(r)\cap\Pi^\pm$, $B_C(r)$ is the open ball of radius $r$ centered at $C$, and $\Pi^\pm: …
- 128k
0
votes
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Another way to do this (again assuming that $\Omega$ is open and connected):
$$\nabla\frac{w_1}{w_2}=\frac{w_1\nabla w_2-w_2\nabla w_1}{w_2^2}
=\frac{w_1}{w_2}\Big(\frac{\nabla w_2}{w_2}-\frac{\nabla …
- 128k
5
votes
Accepted
Intriguing simple question about Sobolev space $W^{1,p}(\Omega)$
Assume that $\Omega$ is open and connected (if $\Omega$ is not connected, then the desired conclusion is clearly false). By convolution with a mollifier and approximation, we may assume that $w_1$ and …
- 128k
4
votes
Solving $X$ for prescribed $\operatorname{div}(X)$ of compact support
This is impossible in general. Indeed, if $X$ has a compact support, then the flux of $X$ through a large enough sphere is $0$, whereas the integral of $f$ over the corresponding ball (say $B$) will n …
- 128k
2
votes
Accepted
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \...
$\newcommand{\R}{\mathbb R}\newcommand{\Z}{\mathbb Z}\newcommand{\de}{\delta}\newcommand{\al}{\alpha}$The answer is no. In fact, $I_t$ does not have to be small even for small $t>0$.
Indeed, take any …
- 128k
3
votes
Accepted
Nontrivial invariant transformations for heat equations
At least when $n=1$, there are no nontrivial transformations of this kind.
Indeed, suppose that $v(t,x)=u(\tau(t,x),\xi(t,x))$, where $u_t=u_{xx}$. Then $u_{tx}=u_{xxx}$ and $u_{tt}=u_{xxxx}$, so that …
- 128k
2
votes
Accepted
Singular integral bounded by Dirichlet form?
$\newcommand{\Om}{\Omega}$The answer is no.
Indeed, by rescaling, without loss of generality $L=1$. Suppose that
\begin{equation}
f(x_1,x_2)=g_a(x_1-x_2)
\end{equation}
for $(x_1,x_2)\in\Om$,
where $ …
- 128k
3
votes
Accepted
Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$
We have
$$\nabla^2 g(t,x)=g(t,x)\Big(\frac{xx^\top}{(2t)^2}-\frac{I_d}{2t}\Big),$$
where $I_d\in\mathbb R^{d\times d}$ is the identity matrix and $\mathbb R^d$ is identified with $\mathbb R^{d\times1} …
- 128k
4
votes
Accepted
Does this dyadic sum converge?
$\newcommand\ep\epsilon\newcommand\Ga\Gamma\newcommand\Z{\mathbb Z}$The answer is yes.
Indeed, let $b:=d/p$, so that $1>a>b>0$. For real $\ep>0$, let
$$K(\ep):=\int_0^\infty e^{-\ep s}\frac{s^a}{1+s^{ …
- 128k
1
vote
Accepted
An estimate of the integral of the higher order derivative of a bump function
$\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\na}{\nabla}$Note that
\begin{equation}
(\na^n\rho_\ep)(x)=\ep^{-d-n}(\na^n\rho)(\ep^{-1}x).
\end{equation}
So,
\begin{equation …
- 128k
2
votes
Accepted
Convolution with the Jacobi Theta-function on "both the space and time variables" - still jo...
$\newcommand\Th\Theta\newcommand{\Z}{\mathbb Z}$The answer here is no.
E.g., for $(x,t)\in[0,1]\times[0,\infty)$ let
\begin{equation}
f(x,t):=2x\,1(x<1/2)+(2-2x)\,1(x\ge1/2).
\end{equation}
Then
…
- 128k
8
votes
Accepted
Asymptotic behavior of a certain oscillatory integral
We can evaluate $I(x)$ explicitly, and then asymptotically.
Indeed, using the substitution $s=ru/x$, we get
\begin{equation*}
I(x)=\frac1{\sqrt x}\lim_{R\to\infty}J_R(x), \tag{1}\label{1}
\end{equ …
- 128k
3
votes
Accepted
Are there $f,h$ such that $h$ is Lipschitz, $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\parti...
This is impossible for any positive function $h$. Indeed, for $|x|=\sqrt t$, the inequality $|\partial_t g| (t, x) \le f(t)g(h(t), x)$ implies
$$f(t)\ge\frac Ct\,u^{-d/2}e^{cu}\ge\frac Bt$$
for all re …
- 128k